MathDB
radicals of distances and heights in tetrahedron inequality

Source: Kyiv City Olympiad 2003 11.3

June 27, 2021
geometry3D geometrygeometrical inequalitytetrahedroninequalities

Problem Statement

Let x1,x2,x3,x4x_1, x_2, x_3, x_4 be the distances from an arbitrary point inside the tetrahedron to the planes of its faces, and let h1,h2,h3,h4h_1, h_2, h_3, h_4 be the corresponding heights of the tetrahedron. Prove that h1+h2+h3+h4≥x1+x2+x3+x4\sqrt{h_1+h_2+h_3+h_4} \ge \sqrt{x_1}+\sqrt{x_2}+\sqrt{x_3}+\sqrt{x_4}
(Dmitry Nomirovsky)
Back to ProblemsView on AoPS